I contacted an Apple Tech Support guy, and he said they have been having log in problems today (Fri, 12 April). He suggested I give it a few hours, and try again. It is not an ongoing site or iTunes problem, so if you were having log-in problems before, it may be another issue. In fact, it could be another issue for me too, but we'll see.
(You can verify this log-in problem here: https://appleid.apple.com/. It's down for me)
I'll update it with news later.